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Mirrors > Home > MPE Home > Th. List > spw | Structured version Visualization version GIF version |
Description: Weak version of the specialization scheme sp 2172. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2172 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2172 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2130 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2172 are spfw 2031 (minimal distinct variable requirements), spnfw 1975 (when 𝑥 is not free in ¬ 𝜑), spvw 1976 (when 𝑥 does not appear in 𝜑), sptruw 1798 (when 𝜑 is true), and spfalw 1995 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |
Ref | Expression |
---|---|
spw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spw | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1902 | . 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
2 | ax-5 1902 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
3 | ax-5 1902 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
4 | spw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 1, 2, 3, 4 | spfw 2031 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 |
This theorem is referenced by: hba1w 2045 spaev 2048 ax12w 2128 bj-ssblem1 33884 bj-ax12w 33907 |
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